1. Field of the Invention
The present invention generally relates to a system and method for reconstructing cone-beam tomographic images and particularly to an iterative process for performing such functionality.
2. Description of the Prior Art
Nondestructive analysis and visualization of three-dimensional microstructures of opaque specimens are important techniques in biomedical and material sciences and engineering. Due to its penetration ability and contrast mechanism, X-ray microtomography is a powerful tool in this type of application. For example, and as described in G. Wang et al., "Cone-beam X-ray microtomography", Multi-dimensional microscopy, Springer-Verlag, New York, Chapter 9, pages 151-169 (1994), an X-ray shadow projection microscope with a microtomography capability is being developed at the State University of New York at Buffalo (SUNY/Buffalo). FIG. 1, described in further detail later, shows such a system.
Cone-beam geometry is also important for emission tomography, medical and industrial X-ray CT. Due to 3D divergency, reconstruction from cone-beam data is much more intricate than in parallel-beam or fan-beam geometry. Having been studied for many years, cone-beam tomography remains a major topic in CT.
A cone-beam reconstruction formula for an infinitely long scanning line has been developed. Additionally, a formula has been derived for reconstruction of a real function with a compact support under the condition that almost every plane through the support meets a scanning locus transversely. A cone-beam reconstruction formula for two intersecting source circles has also been established. Thorough theoretical analyses on cone-beam reconstruction have been reported by Tuy, Smith and Grangeat. As a result, the following sufficient condition for exact cone-beam reconstruction has been developed:
if on every plane through an object there exists at least one source point, exact cone-beam reconstruction can be achieved. PA1 if a source curve is connected and compact, and if its convex hull contains a planar region, the cone-beam data from the source curve is complete for exact reconstruction of the planar region.
A theoretical framework for local cone-beam reconstruction has been developed. The concept of the planar region of an object as intersection of the object support and a set of planes has been defined, proving that
An estimation formula for local reconstruction and a convergence condition for the formula have also been created.
Exact cone-beam reconstruction algorithms have been implemented in the past, including filtered backprojection algorithms. Among current exact algorithms, a direct Fourier transform method is computationally the most efficient for a sufficiently large amount of data.
Approximate cone-beam reconstruction algorithms are also important in practice. Generally speaking, approximate cone-beam reconstruction cannot be avoided in the cases of incomplete scanning geometry and partial detection coverage. Furthermore, approximate reconstruction is usually associated with higher computational efficiency, and likely less image noise and ringing. Others have adapted the equispatial fan-beam algorithm for cone-beam reconstruction with a circular scanning locus. Because this algorithm is limited by circular scanning, spherical specimen support and longitudinal image blurring, it has been extended in various ways.
Tam, U.S. Pat. No. 5,270,926, discloses an iterative algorithm for cone-beam reconstruction from incomplete data. In this algorithm, Radon data are first computed, and missing data initialized to zero. Then, an image volume is reconstructed slice-by-slice via 2D filtered backprojection. Projection data are corrected by a priori information on the object support, upper and lower bounds of projection values, and reprojected to calculate the missing data. The steps are repeated until some convergence criterion is satisfied.
However, the convergence and optimality of this iterative algorithm have not been established. Other existing cone-beam algorithms require that projections be complete at least along one direction, and therefore cone-beam reconstruction is impossible in cases where objects contain X-ray opaque components and/or are larger than the cone-beam aperture defined by effective detection area and X-ray source position.
The expectation maximization (EM) approach is well known. This iterative formula has been interpreted in a deterministic sense, and its desirable theoretical properties have been established. Based on this formula, 2-D parallel-beam CT algorithms for metal artifact reduction and local reconstruction from truncated data may be developed.